Psychology of Mathematics for Instruction

L. B. Resnick, W. W. Ford

• Publisher: Routledge
• Year: 2012
• Print ISBN:
  9780898590296
• eBook ISBN:
  9781136557590
• Topic: Pedagogía
• Subtopic: Educación general

Published in 1981, Psychology of Mathematics for Instruction is a valuable contribution to the field of Education.

I	MATHEMATICS AS COMPUTATION

One way to define mathematics is as a body of computational rules and procedures. To the layman such a definition seems quite natural and, in fact, this definition pervades most of mathematics instruction in our elementary schools. If one asks a 10-year-old girl, “What are you learning in math these days?”, she is far more likely to mention long division, multiplication tables, or addition of fractions than she is to mention sets, commutativity, inequalities, and functions. Elementary school mathematics is dominated by computation, and computational proficiency remains a major goal for instruction, despite efforts to reform the mathematics curriculum in the 1960s. The aim of the reform movement was to introduce into mathematics instruction as early as possible certain basic concepts of the discipline, such as the properties of the real-number system, simple geometric constructs, logic, and set theory (see Chapter 5). The reformers thought that if sufficient time and thought were devoted to teaching children the underlying constructs of mathematics, computational skill would follow along rather nicely. They assumed that computational proficiency could be developed without devoting a lot of time to computation in a direct way.

Curriculum changes notwithstanding, there is no doubt that computation still constitutes the bulk of children's experiences in elementary school mathematics. We expect children to know how to perform complex computations and to do them quickly and accurately. We also aim, but less successfully, at making them capable of applying these computational skills in problem solving.

Let us look briefly at the computational skills themselves. By computation, we mean addition, subtraction, multiplication, and division. We also mean the use of percents, fractions, and certain other basic, everyday-life kinds of skills. We mean, in short, what has traditionally been called arithmetic. In thinking about arithmetic or computational skills, it is useful to distinguish between simple associations, sometimes called the number facts, and complex procedures, called algorithms, in which a fixed sequence of operations has to be performed. The simple associations or number facts are the tables—the addition, subtraction, multiplication, and sometimes division tables—that most children are expected to memorize in elementary school. The complex procedures or algorithms are the series of separate steps required, for example, in performing long division or subtracting fractions with different denominators. Individuals executing such procedures must know what steps to perform, must perform them in the proper order, and must recall needed number facts accurately. When children learn to use number facts and algorithms in problem solving, they typically do so in the context of verbally stated problems, known as word problems or story problems. Word problems require children to interpret the words of the problem, set up an equivalent mathematical calculation, and then apply relevant procedures.

In the next three chapters, we consider what psychology offers to the instructor whose goal is to develop these kinds of computational skills in children. We look first at the historical and current uses of drill and practice as a means of developing speed and accuracy in computation. We then show how computational tasks have been analyzed into component skills using the method of hierarchy generation, and we suggest how informal task analysis may be used by teachers to plan the organization of instruction. The final chapter in Part I introduces a variety of methods for studying the complex mental processing that goes on while children are carrying out computational procedures.

2	The Psychology Of Drill and Practice

When we think back on our own school days, to the hours we spent on arithmetic, many of us remember laboring over pages of problems. Often these were pages of identical calculations, where only the numbers were varied. Or we worked with flash cards until we could shout out the answers immediately and with no mistakes. This kind of work was called “drill and practice.” It was supposed to help us achieve perfect mastery of basic addition, subtraction, multiplication, and division. It was to ensure that we would forever remember how to perform the arithmetic operations we had been taught.

Drill and practice has a place of long standing in the history of mathematics teaching, especially in arithmetic. At one time it was the major means of instruction. Today it is still part of the mathematics curriculum, although usually accompanied by concrete experiences or explanations of underlying mathematical principles. Most everyone accepts some form of practice as necessary. The reason, according to educators and lay people alike, is that “practice makes perfect.” Along with drill and practice come increases in speed and accuracy, which are two widely accepted criteria of computational proficiency. If children can execute calculations speedily and accurately, most people are satisfied that they “know” their computational skills.

What do psychologists know about the role of drill in establishing and maintaining computational proficiency? This chapter explores the historical and theoretical bases for including drill and practice in the mathematics curriculum. We begin with a look at a psychological theory-associationism-that provides one theoretical justification for the use of drill exercises. We have chosen to focus on E. L. Thorndike, who is, in a sense, the “founding father” of the psychology of mathematics instruction. As a psychologist Thorndike was firmly rooted in a tradition of laboratory experimentalism; but he was also strongly committed to the task of translating laboratory findings into guidelines for classroom instruction. We also present the arguments advanced against drill methods by another psychologist, William Brownell. Because of its prominence in arithmetic teaching, practice has received a great deal of study, especially during the first half of this century, most of it geared to make drill better organized and more effective. We survey a line of research that attempted to determine the relative ease or difficulty of arithmetic problems—and to account for those differences—so that teachers could plan the proper amounts and sequences of practice. We describe a computer-assisted drill program as an example of one way psychologists have attempted to optimize amounts of practice and rates of progression through drill material. Finally, we present theory and research that indicate why it might be important to develop speed and accuracy in certain kinds of computations.

EDWARD L. THORNDIKE AND THE FORMATION OF BONDS

In 1922 a small book appeared, called The Psychology of Arithmetic. It was written by Edward L. Thorndike, a psychologist working at Teachers College of Columbia University, who helped to develop some early principles of stimulus-response learning psychology. Thorndike is perhaps best known in psychology for his statement of the law of effect, an early version of what we now call principles of reinforcement. He discovered this law, not in the context of a complex subject like mathematics but in the context of simple laboratory experiments with cats, dogs, monkeys, and chickens.

According to Thorndike, in any given situation an animal had a number of possible responses, and the action that would be performed depended on the strength of the “connection” or “bond” between the situation and the specific action. The experiment most frequently associated with this idea involved placing a cat in a wooden puzzle box that could be opened by tripping a latch. Naturally, the cat would object to being confined in such close quarters and would claw and scratch at the side of the box to get out. Eventually it would accidentally trip the latch, opening the door and escaping. Replaced in the box, the cat would again claw and scratch; but each time the experiment was repeated, the cat took less time to find its way out. Of all the clawing and scratching responses, only the one that opened the door was rewarded by the opportunity for escape. In Thorndike's conception, the cat was not “figuring out” how to open the box; rather the reward of escape was serving to strengthen the bonds between the experimental situation and the particular response that permitted escape. Hence Thorndike's formulation of the law of effect (Thorndike, 1913): “When a modifiable connection between a situation and a response is made and is accompanied or followed by a satisfying state of affairs, that connection's strength is increased: When made and accompanied or followed by an annoying state of affairs, its strength is decreased [p. 4].”

Though he experimented mostly with animals, Thorndike thought his learning principles should apply equally to humans. Along with many other psychologists of the time—called “connectionists” or “associationists”—Thorndike argued that all human behavior, thought as well as action, could be analyzed in terms of two simple constructs. When broken down into its most basic units, behavior would be found to consist of stimuli, or events external to the person, and responses, or things that people did in reaction to those external events. When a certain response was given to a certain stimulus and followed by a reward, then a bond, or association, began to be formed between the stimulus and the response. The more frequently a certain stimulus–response pair was rewarded, the stronger the bond. Thus the law of effect—a special case of the laws of association—suggested that practice followed by reward was an important way in which human learning took place.

Associations between stimuli and responses, bonds, and the law of effect—how could these principles, developed largely by observing animals perform the simplest of behaviors, be applied to something as complex as school learning? That was the question that Thorndike (1922) addressed in The Psychology of Arithmetic. The answer seemed straightforward because associationism held that all knowledge, even the most complex, was built of these simple connections. Learning thus consisted of establishing and strengthening the needed associations. “The aims of elementary education,” Thorndike said, “when fully defined, will be found to be the production of changes in human nature represented by an almost countless list of connections or bonds whereby the pupil thinks or feels or acts in certain ways in response to the situations the school has organized and is influenced to think and feel and act similarly to similar situations when life outside of school confronts him with them [p. xi].”

Rather than simply announcing the laws of learning to teachers and educators, Thorndike set out to demonstrate how they could be applied to the problems of instruction. What teachers needed, he believed, was to find and make explicit the particular set of bonds that constituted arithmetic. Once well-organized lists of all these bonds could be drawn up, then rewarded practice would enable the law of effect to strengthen these bonds, and one could expect improved performance in arithmetic. Thorndike's book was an attempt to explain how the subject matter of arithmetic could be translated into psychologically formulated stimulus–response bonds.

Because children of elementary school age were not yet able to deduce the rules of arithmetic from examples and other rules, Thorndike reasoned, the task of instruction was to form carefully the necessary bonds and habits that would allow them to perform computations and solve problems. As a first step, one would have to select the bonds to be formed. Naturally, any carefully constructed arithmetic curriculum, with or without benefit of psychological analysis, would divide the subject matter up into broadly defined topics. For example, multiplication would be treated as a composite of abilities, such as: “knowledge of multiplication tables up to 9 × 9; ability to multiply two (or more) place numbers when carrying is not required and no zeros occur in the multiplicand; ability to multiply by 2, 3,…, 9, with carrying;” and so forth, up to the ability to multiply two-place decimals (as with United States money), with fractions, and with mixed numbers. What Thorndike, as a psychologist, proposed was to analyze these abilities further into a detailed set of mental habits or connections, each of which would become a candidate for formation and strengthening. Figure 2.1 shows an analysis of simple addition in columns, of which Thorndike (1922) says: “The majority of teachers probably treat this as a simple application of the knowledge of the additions to 9 + 9, plus understanding of ‘carrying.’ On the contrary there are at least seven processes or minor functions involved in two-place column addition, each of which is psychologically distinct and requires distinct educational treatment [p. 52].”

Once the proper bonds were selected, how could they be formed and strengthened? This was where drill and practice came in. Proper drill and practice, according to Thorndike, involved presenting bonds in a carefully programmed way so that important bonds were practiced often, and lesser bonds, less often. So-called “propaedeutic” bonds, used only to facilitate learning new concepts, would be practiced temporarily but later drop out from disuse. For example, to add four 5’s in a column, a child might be taught a propaedeutic bond like counting 5, 10, 15, 20; however, because this was to be replaced later by the bond “four 5’s are 20,” it would receive only minimal practice. Bonds were recognized to have an effect on each other; hence Thorndike (1922) noted: “Every bond formed should be formed with due consideration of every other bond that has been or will be formed; every ability should be practiced in the most effective possible relations with other abilities [p. 140].” The reward that served to strengthen the practiced bonds was obtained when arithmetic problems were made interesting, fun, and close to practical applications. Thus, Thorndike was also concerned with the intrinsic meaningfulness of problems and their relevance to daily activities outside of school.

FIG. 2.1 Thorndike's analysis of column addition into bonds. (From Thorndike, 1922.)

Some of Thorndike's bonds seem quite straightforward. It is easy for us to imagine that learning a bond like “2 + 2” (the stimulus) equals “4” (the response) would be enhanced by appropriate forms of drill. Arithmetic is full of these simple bonds. But not all arithmetic is so easy to translate into stimulus-response terms. As anyone knows who has tried to learn long division, some of it involves extremely long and complex operations. Thorndike (1922) explained these complex operations as “organized cooperating system(s) of bonds [p. 138],” groups of individual bonds that needed to be taught “teamwork” by being carefully sequenced and practiced, as described in the following quotation:

As each new ability is acquired, then, we seek to have it take its place as an improvement of a thinking being, as a cooperative member of a total organization, as a soldier fighting together with others, as an element in an educated personality. Such an organization of bonds will not form itself any more than any one bond will create itself. If the elements of arithmetical ability are to act together as a total organized unified force, they must be made to act together in the course of learning. What w...

Table of contents

• Front Cover
• Title Page
• Copyright
• CONTENTS
• Preface
• INTRODUCTION
• PART I: MATHEMATICS AS COMPUTATION
• PART II: MATHEMATICS AS CONCEPTUAL UNDERSTANDING AND PROBLEM SOLVING
• CONCLUSION
• Author Index
• Subject Index